Oscar Zariski

Oscar Zariski
Oscar Zariski (1899–1986)
Born	(1899-04-24)April 24, 1899 Kobrin, Russian Empire
Died	July 4, 1986(1986-07-04) (aged 87) Brookline, Massachusetts, United States
Nationality	American
Fields	Mathematics
Institutions	Johns Hopkins University University of Illinois Harvard University
Alma mater	University of Rome University of Kiev
Doctoral advisor	Guido Castelnuovo
Doctoral students	S. S. Abhyankar Michael Artin Iacopo Barsotti Irvin Cohen Daniel Gorenstein Robin Hartshorne Heisuke Hironaka Steven Kleiman Joseph Lipman David Mumford Maxwell Rosenlicht Pierre Samuel Abraham Seidenberg
Known for	Contributions to algebraic geometry
Notable awards	Cole Prize in Algebra (1944) National Medal of Science (1965) Wolf Prize in Mathematics (1981) Steele Prize

Oscar Zariski (born Oscher Zaritsky (Russian: О́скар Зари́сский; April 24, 1899 – July 4, 1986) was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kiev. He left Kiev in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi.

Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski.

Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school. The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference.

It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively).